Multiparameter quantum metrology with postselection measurements

We analyze simultaneous quantum estimations of multiple parameters with postselection measurements in terms of a trade-off relation. The system, or a sensor, is characterized by a set of parameters, interacts with a measurement apparatus (MA), and then is postselected onto a set of orthonormal final states. Measurements of the MA yield an estimation of the parameters. We first derive classical and quantum Cramér-Rao lower bounds and then discuss their archivable condition and the trade-offs in the postselection measurements, in general, including the case when a sensor is in a mixed state. Its whole information can, in principle, be obtained via the MA, which is not possible without postselection. We then apply the framework to simultaneous measurements of phase and its fluctuation as an example.